LMFDB - Embedded newform 1850.2.b.h.149.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1850,2,Mod(149,1850)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1850, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1850.149");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1850 = 2 \cdot 5^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1850.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$14.7723243739$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			149.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1850.149
Dual form			1850.2.b.h.149.2

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} +2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +4.00000 q^{11} -2.00000i q^{12} -2.00000i q^{13} +1.00000 q^{16} -8.00000i q^{17} +1.00000i q^{18} +5.00000 q^{19} -4.00000i q^{22} +1.00000i q^{23} -2.00000 q^{24} -2.00000 q^{26} +4.00000i q^{27} -10.0000 q^{29} -4.00000 q^{31} -1.00000i q^{32} +8.00000i q^{33} -8.00000 q^{34} +1.00000 q^{36} -1.00000i q^{37} -5.00000i q^{38} +4.00000 q^{39} +7.00000 q^{41} -9.00000i q^{43} -4.00000 q^{44} +1.00000 q^{46} -6.00000i q^{47} +2.00000i q^{48} +7.00000 q^{49} +16.0000 q^{51} +2.00000i q^{52} -3.00000i q^{53} +4.00000 q^{54} +10.0000i q^{57} +10.0000i q^{58} +11.0000 q^{59} +2.00000 q^{61} +4.00000i q^{62} -1.00000 q^{64} +8.00000 q^{66} -2.00000i q^{67} +8.00000i q^{68} -2.00000 q^{69} +14.0000 q^{71} -1.00000i q^{72} +3.00000i q^{73} -1.00000 q^{74} -5.00000 q^{76} -4.00000i q^{78} +11.0000 q^{79} -11.0000 q^{81} -7.00000i q^{82} +8.00000i q^{83} -9.00000 q^{86} -20.0000i q^{87} +4.00000i q^{88} +2.00000 q^{89} -1.00000i q^{92} -8.00000i q^{93} -6.00000 q^{94} +2.00000 q^{96} +8.00000i q^{97} -7.00000i q^{98} -4.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 4 q^{6} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 + 4 * q^6 - 2 * q^9	$ 2 q - 2 q^{4} + 4 q^{6} - 2 q^{9} + 8 q^{11} + 2 q^{16} + 10 q^{19} - 4 q^{24} - 4 q^{26} - 20 q^{29} - 8 q^{31} - 16 q^{34} + 2 q^{36} + 8 q^{39} + 14 q^{41} - 8 q^{44} + 2 q^{46} + 14 q^{49} + 32 q^{51} + 8 q^{54} + 22 q^{59} + 4 q^{61} - 2 q^{64} + 16 q^{66} - 4 q^{69} + 28 q^{71} - 2 q^{74} - 10 q^{76} + 22 q^{79} - 22 q^{81} - 18 q^{86} + 4 q^{89} - 12 q^{94} + 4 q^{96} - 8 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 + 4 * q^6 - 2 * q^9 + 8 * q^11 + 2 * q^16 + 10 * q^19 - 4 * q^24 - 4 * q^26 - 20 * q^29 - 8 * q^31 - 16 * q^34 + 2 * q^36 + 8 * q^39 + 14 * q^41 - 8 * q^44 + 2 * q^46 + 14 * q^49 + 32 * q^51 + 8 * q^54 + 22 * q^59 + 4 * q^61 - 2 * q^64 + 16 * q^66 - 4 * q^69 + 28 * q^71 - 2 * q^74 - 10 * q^76 + 22 * q^79 - 22 * q^81 - 18 * q^86 + 4 * q^89 - 12 * q^94 + 4 * q^96 - 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1850\mathbb{Z}\right)^\times$.

$n$	$1001$	$1777$
$\chi(n)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	2.00000i	1.15470i	0.816497	+	0.577350i	$0.195913\pi$
$3$	2.00000i	1.15470i	−0.816497	+	0.577350i	$0.804087\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	2.00000	0.816497
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	1.00000i	0.353553i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	− 2.00000i	− 0.577350i
$13$	− 2.00000i	− 0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$	− 2.00000i	− 0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	− 8.00000i	− 1.94029i	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	− 8.00000i	− 1.94029i	0.242536	−	0.970143i	$-0.422021\pi$
$18$	1.00000i	0.235702i
$19$	5.00000	1.14708	0.573539	−	0.819178i	$-0.305570\pi$
$19$	5.00000	1.14708	0.573539	+	0.819178i	$0.305570\pi$
$20$	0	0
$21$	0	0
$22$	− 4.00000i	− 0.852803i
$23$	1.00000i	0.208514i	0.994550	+	0.104257i	$0.0332465\pi$
$23$	1.00000i	0.208514i	−0.994550	+	0.104257i	$0.966753\pi$
$24$	−2.00000	−0.408248
$25$	0	0
$26$	−2.00000	−0.392232
$27$	4.00000i	0.769800i
$28$	0	0
$29$	−10.0000	−1.85695	−0.928477	−	0.371391i	$-0.878881\pi$
$29$	−10.0000	−1.85695	−0.928477	+	0.371391i	$0.878881\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	− 1.00000i	− 0.176777i
$33$	8.00000i	1.39262i
$34$	−8.00000	−1.37199
$35$	0	0
$36$	1.00000	0.166667
$37$	− 1.00000i	− 0.164399i
$37$	− 1.00000i	− 0.164399i
$38$	− 5.00000i	− 0.811107i
$39$	4.00000	0.640513
$40$	0	0
$41$	7.00000	1.09322	0.546608	−	0.837389i	$-0.315919\pi$
$41$	7.00000	1.09322	0.546608	+	0.837389i	$0.315919\pi$
$42$	0	0
$43$	− 9.00000i	− 1.37249i	−0.727372	−	0.686244i	$-0.759258\pi$
$43$	− 9.00000i	− 1.37249i	0.727372	−	0.686244i	$-0.240742\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	1.00000	0.147442
$47$	− 6.00000i	− 0.875190i	−0.899172	−	0.437595i	$-0.855830\pi$
$47$	− 6.00000i	− 0.875190i	0.899172	−	0.437595i	$-0.144170\pi$
$48$	2.00000i	0.288675i
$49$	7.00000	1.00000
$50$	0	0
$51$	16.0000	2.24045
$52$	2.00000i	0.277350i
$53$	− 3.00000i	− 0.412082i	−0.978543	−	0.206041i	$-0.933942\pi$
$53$	− 3.00000i	− 0.412082i	0.978543	−	0.206041i	$-0.0660580\pi$
$54$	4.00000	0.544331
$55$	0	0
$56$	0	0
$57$	10.0000i	1.32453i
$58$	10.0000i	1.31306i
$59$	11.0000	1.43208	0.716039	−	0.698060i	$-0.245953\pi$
$59$	11.0000	1.43208	0.716039	+	0.698060i	$0.245953\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	4.00000i	0.508001i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	8.00000	0.984732
$67$	− 2.00000i	− 0.244339i	−0.992509	−	0.122169i	$-0.961015\pi$
$67$	− 2.00000i	− 0.244339i	0.992509	−	0.122169i	$-0.0389851\pi$
$68$	8.00000i	0.970143i
$69$	−2.00000	−0.240772
$70$	0	0
$71$	14.0000	1.66149	0.830747	−	0.556650i	$-0.187914\pi$
$71$	14.0000	1.66149	0.830747	+	0.556650i	$0.187914\pi$
$72$	− 1.00000i	− 0.117851i
$73$	3.00000i	0.351123i	0.984468	+	0.175562i	$0.0561742\pi$
$73$	3.00000i	0.351123i	−0.984468	+	0.175562i	$0.943826\pi$
$74$	−1.00000	−0.116248
$75$	0	0
$76$	−5.00000	−0.573539
$77$	0	0
$78$	− 4.00000i	− 0.452911i
$79$	11.0000	1.23760	0.618798	−	0.785550i	$-0.287620\pi$
$79$	11.0000	1.23760	0.618798	+	0.785550i	$0.287620\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	− 7.00000i	− 0.773021i
$83$	8.00000i	0.878114i	0.898459	+	0.439057i	$0.144687\pi$
$83$	8.00000i	0.878114i	−0.898459	+	0.439057i	$0.855313\pi$
$84$	0	0
$85$	0	0
$86$	−9.00000	−0.970495
$87$	− 20.0000i	− 2.14423i
$88$	4.00000i	0.426401i
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	0	0
$92$	− 1.00000i	− 0.104257i
$93$	− 8.00000i	− 0.829561i
$94$	−6.00000	−0.618853
$95$	0	0
$96$	2.00000	0.204124
$97$	8.00000i	0.812277i	0.913812	+	0.406138i	$0.133125\pi$
$97$	8.00000i	0.812277i	−0.913812	+	0.406138i	$0.866875\pi$
$98$	− 7.00000i	− 0.707107i
$99$	−4.00000	−0.402015
$100$	0	0
$101$	−9.00000	−0.895533	−0.447767	−	0.894150i	$-0.647781\pi$
$101$	−9.00000	−0.895533	−0.447767	+	0.894150i	$0.647781\pi$
$102$	− 16.0000i	− 1.58424i
$103$	17.0000i	1.67506i	0.546392	+	0.837530i	$0.316001\pi$
$103$	17.0000i	1.67506i	−0.546392	+	0.837530i	$0.683999\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	−3.00000	−0.291386
$107$	10.0000i	0.966736i	0.875417	+	0.483368i	$0.160587\pi$
$107$	10.0000i	0.966736i	−0.875417	+	0.483368i	$0.839413\pi$
$108$	− 4.00000i	− 0.384900i
$109$	16.0000	1.53252	0.766261	−	0.642529i	$-0.222115\pi$
$109$	16.0000	1.53252	0.766261	+	0.642529i	$0.222115\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	0	0
$113$	− 2.00000i	− 0.188144i	−0.995565	−	0.0940721i	$-0.970012\pi$
$113$	− 2.00000i	− 0.188144i	0.995565	−	0.0940721i	$-0.0299884\pi$
$114$	10.0000	0.936586
$115$	0	0
$116$	10.0000	0.928477
$117$	2.00000i	0.184900i
$118$	− 11.0000i	− 1.01263i
$119$	0	0
$120$	0	0
$121$	5.00000	0.454545
$122$	− 2.00000i	− 0.181071i
$123$	14.0000i	1.26234i
$124$	4.00000	0.359211
$125$	0	0
$126$	0	0
$127$	− 4.00000i	− 0.354943i	−0.984126	−	0.177471i	$-0.943208\pi$
$127$	− 4.00000i	− 0.354943i	0.984126	−	0.177471i	$-0.0567917\pi$
$128$	1.00000i	0.0883883i
$129$	18.0000	1.58481
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	− 8.00000i	− 0.696311i
$133$	0	0
$134$	−2.00000	−0.172774
$135$	0	0
$136$	8.00000	0.685994
$137$	10.0000i	0.854358i	0.904167	+	0.427179i	$0.140493\pi$
$137$	10.0000i	0.854358i	−0.904167	+	0.427179i	$0.859507\pi$
$138$	2.00000i	0.170251i
$139$	6.00000	0.508913	0.254457	−	0.967084i	$-0.418103\pi$
$139$	6.00000	0.508913	0.254457	+	0.967084i	$0.418103\pi$
$140$	0	0
$141$	12.0000	1.01058
$142$	− 14.0000i	− 1.17485i
$143$	− 8.00000i	− 0.668994i
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	3.00000	0.248282
$147$	14.0000i	1.15470i
$148$	1.00000i	0.0821995i
$149$	−3.00000	−0.245770	−0.122885	−	0.992421i	$-0.539215\pi$
$149$	−3.00000	−0.245770	−0.122885	+	0.992421i	$0.539215\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	5.00000i	0.405554i
$153$	8.00000i	0.646762i
$154$	0	0
$155$	0	0
$156$	−4.00000	−0.320256
$157$	− 7.00000i	− 0.558661i	−0.960195	−	0.279330i	$-0.909888\pi$
$157$	− 7.00000i	− 0.558661i	0.960195	−	0.279330i	$-0.0901125\pi$
$158$	− 11.0000i	− 0.875113i
$159$	6.00000	0.475831
$160$	0	0
$161$	0	0
$162$	11.0000i	0.864242i
$163$	− 9.00000i	− 0.704934i	−0.935824	−	0.352467i	$-0.885343\pi$
$163$	− 9.00000i	− 0.704934i	0.935824	−	0.352467i	$-0.114657\pi$
$164$	−7.00000	−0.546608
$165$	0	0
$166$	8.00000	0.620920
$167$	− 23.0000i	− 1.77979i	−0.456162	−	0.889897i	$-0.650776\pi$
$167$	− 23.0000i	− 1.77979i	0.456162	−	0.889897i	$-0.349224\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	−5.00000	−0.382360
$172$	9.00000i	0.686244i
$173$	− 2.00000i	− 0.152057i	−0.997106	−	0.0760286i	$-0.975776\pi$
$173$	− 2.00000i	− 0.152057i	0.997106	−	0.0760286i	$-0.0242240\pi$
$174$	−20.0000	−1.51620
$175$	0	0
$176$	4.00000	0.301511
$177$	22.0000i	1.65362i
$178$	− 2.00000i	− 0.149906i
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	−7.00000	−0.520306	−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000	−0.520306	−0.260153	+	0.965567i	$0.583773\pi$
$182$	0	0
$183$	4.00000i	0.295689i
$184$	−1.00000	−0.0737210
$185$	0	0
$186$	−8.00000	−0.586588
$187$	− 32.0000i	− 2.34007i
$188$	6.00000i	0.437595i
$189$	0	0
$190$	0	0
$191$	−19.0000	−1.37479	−0.687396	−	0.726283i	$-0.741246\pi$
$191$	−19.0000	−1.37479	−0.687396	+	0.726283i	$0.741246\pi$
$192$	− 2.00000i	− 0.144338i
$193$	20.0000i	1.43963i	0.694165	+	0.719816i	$0.255774\pi$
$193$	20.0000i	1.43963i	−0.694165	+	0.719816i	$0.744226\pi$
$194$	8.00000	0.574367
$195$	0	0
$196$	−7.00000	−0.500000
$197$	− 7.00000i	− 0.498729i	−0.968410	−	0.249365i	$-0.919778\pi$
$197$	− 7.00000i	− 0.498729i	0.968410	−	0.249365i	$-0.0802218\pi$
$198$	4.00000i	0.284268i
$199$	−1.00000	−0.0708881	−0.0354441	−	0.999372i	$-0.511285\pi$
$199$	−1.00000	−0.0708881	−0.0354441	+	0.999372i	$0.511285\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	9.00000i	0.633238i
$203$	0	0
$204$	−16.0000	−1.12022
$205$	0	0
$206$	17.0000	1.18445
$207$	− 1.00000i	− 0.0695048i
$208$	− 2.00000i	− 0.138675i
$209$	20.0000	1.38343
$210$	0	0
$211$	−22.0000	−1.51454	−0.757271	−	0.653101i	$-0.773468\pi$
$211$	−22.0000	−1.51454	−0.757271	+	0.653101i	$0.773468\pi$
$212$	3.00000i	0.206041i
$213$	28.0000i	1.91853i
$214$	10.0000	0.683586
$215$	0	0
$216$	−4.00000	−0.272166
$217$	0	0
$218$	− 16.0000i	− 1.08366i
$219$	−6.00000	−0.405442
$220$	0	0
$221$	−16.0000	−1.07628
$222$	− 2.00000i	− 0.134231i
$223$	6.00000i	0.401790i	0.979613	+	0.200895i	$0.0643850\pi$
$223$	6.00000i	0.401790i	−0.979613	+	0.200895i	$0.935615\pi$
$224$	0	0
$225$	0	0
$226$	−2.00000	−0.133038
$227$	7.00000i	0.464606i	0.972643	+	0.232303i	$0.0746261\pi$
$227$	7.00000i	0.464606i	−0.972643	+	0.232303i	$0.925374\pi$
$228$	− 10.0000i	− 0.662266i
$229$	1.00000	0.0660819	0.0330409	−	0.999454i	$-0.489481\pi$
$229$	1.00000	0.0660819	0.0330409	+	0.999454i	$0.489481\pi$
$230$	0	0
$231$	0	0
$232$	− 10.0000i	− 0.656532i
$233$	− 29.0000i	− 1.89985i	−0.312473	−	0.949927i	$-0.601157\pi$
$233$	− 29.0000i	− 1.89985i	0.312473	−	0.949927i	$-0.398843\pi$
$234$	2.00000	0.130744
$235$	0	0
$236$	−11.0000	−0.716039
$237$	22.0000i	1.42905i
$238$	0	0
$239$	13.0000	0.840900	0.420450	−	0.907316i	$-0.361872\pi$
$239$	13.0000	0.840900	0.420450	+	0.907316i	$0.361872\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	− 5.00000i	− 0.321412i
$243$	− 10.0000i	− 0.641500i
$244$	−2.00000	−0.128037
$245$	0	0
$246$	14.0000	0.892607
$247$	− 10.0000i	− 0.636285i
$248$	− 4.00000i	− 0.254000i
$249$	−16.0000	−1.01396
$250$	0	0
$251$	−21.0000	−1.32551	−0.662754	−	0.748837i	$-0.730613\pi$
$251$	−21.0000	−1.32551	−0.662754	+	0.748837i	$0.730613\pi$
$252$	0	0
$253$	4.00000i	0.251478i
$254$	−4.00000	−0.250982
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 8.00000i	− 0.499026i	−0.968371	−	0.249513i	$-0.919729\pi$
$257$	− 8.00000i	− 0.499026i	0.968371	−	0.249513i	$-0.0802706\pi$
$258$	− 18.0000i	− 1.12063i
$259$	0	0
$260$	0	0
$261$	10.0000	0.618984
$262$	12.0000i	0.741362i
$263$	14.0000i	0.863277i	0.902047	+	0.431638i	$0.142064\pi$
$263$	14.0000i	0.863277i	−0.902047	+	0.431638i	$0.857936\pi$
$264$	−8.00000	−0.492366
$265$	0	0
$266$	0	0
$267$	4.00000i	0.244796i
$268$	2.00000i	0.122169i
$269$	−31.0000	−1.89010	−0.945052	−	0.326921i	$-0.893989\pi$
$269$	−31.0000	−1.89010	−0.945052	+	0.326921i	$0.893989\pi$
$270$	0	0
$271$	−4.00000	−0.242983	−0.121491	−	0.992592i	$-0.538768\pi$
$271$	−4.00000	−0.242983	−0.121491	+	0.992592i	$0.538768\pi$
$272$	− 8.00000i	− 0.485071i
$273$	0	0
$274$	10.0000	0.604122
$275$	0	0
$276$	2.00000	0.120386
$277$	− 10.0000i	− 0.600842i	−0.953807	−	0.300421i	$-0.902873\pi$
$277$	− 10.0000i	− 0.600842i	0.953807	−	0.300421i	$-0.0971271\pi$
$278$	− 6.00000i	− 0.359856i
$279$	4.00000	0.239474
$280$	0	0
$281$	28.0000	1.67034	0.835170	−	0.549992i	$-0.185369\pi$
$281$	28.0000	1.67034	0.835170	+	0.549992i	$0.185369\pi$
$282$	− 12.0000i	− 0.714590i
$283$	3.00000i	0.178331i	0.996017	+	0.0891657i	$0.0284201\pi$
$283$	3.00000i	0.178331i	−0.996017	+	0.0891657i	$0.971580\pi$
$284$	−14.0000	−0.830747
$285$	0	0
$286$	−8.00000	−0.473050
$287$	0	0
$288$	1.00000i	0.0589256i
$289$	−47.0000	−2.76471
$290$	0	0
$291$	−16.0000	−0.937937
$292$	− 3.00000i	− 0.175562i
$293$	9.00000i	0.525786i	0.964825	+	0.262893i	$0.0846766\pi$
$293$	9.00000i	0.525786i	−0.964825	+	0.262893i	$0.915323\pi$
$294$	14.0000	0.816497
$295$	0	0
$296$	1.00000	0.0581238
$297$	16.0000i	0.928414i
$298$	3.00000i	0.173785i
$299$	2.00000	0.115663
$300$	0	0
$301$	0	0
$302$	8.00000i	0.460348i
$303$	− 18.0000i	− 1.03407i
$304$	5.00000	0.286770
$305$	0	0
$306$	8.00000	0.457330
$307$	12.0000i	0.684876i	0.939540	+	0.342438i	$0.111253\pi$
$307$	12.0000i	0.684876i	−0.939540	+	0.342438i	$0.888747\pi$
$308$	0	0
$309$	−34.0000	−1.93419
$310$	0	0
$311$	21.0000	1.19080	0.595400	−	0.803429i	$-0.296993\pi$
$311$	21.0000	1.19080	0.595400	+	0.803429i	$0.296993\pi$
$312$	4.00000i	0.226455i
$313$	− 10.0000i	− 0.565233i	−0.959233	−	0.282617i	$-0.908798\pi$
$313$	− 10.0000i	− 0.565233i	0.959233	−	0.282617i	$-0.0912024\pi$
$314$	−7.00000	−0.395033
$315$	0	0
$316$	−11.0000	−0.618798
$317$	27.0000i	1.51647i	0.651981	+	0.758236i	$0.273938\pi$
$317$	27.0000i	1.51647i	−0.651981	+	0.758236i	$0.726062\pi$
$318$	− 6.00000i	− 0.336463i
$319$	−40.0000	−2.23957
$320$	0	0
$321$	−20.0000	−1.11629
$322$	0	0
$323$	− 40.0000i	− 2.22566i
$324$	11.0000	0.611111
$325$	0	0
$326$	−9.00000	−0.498464
$327$	32.0000i	1.76960i
$328$	7.00000i	0.386510i
$329$	0	0
$330$	0	0
$331$	−24.0000	−1.31916	−0.659580	−	0.751635i	$-0.729266\pi$
$331$	−24.0000	−1.31916	−0.659580	+	0.751635i	$0.729266\pi$
$332$	− 8.00000i	− 0.439057i
$333$	1.00000i	0.0547997i
$334$	−23.0000	−1.25850
$335$	0	0
$336$	0	0
$337$	1.00000i	0.0544735i	0.999629	+	0.0272367i	$0.00867079\pi$
$337$	1.00000i	0.0544735i	−0.999629	+	0.0272367i	$0.991329\pi$
$338$	− 9.00000i	− 0.489535i
$339$	4.00000	0.217250
$340$	0	0
$341$	−16.0000	−0.866449
$342$	5.00000i	0.270369i
$343$	0	0
$344$	9.00000	0.485247
$345$	0	0
$346$	−2.00000	−0.107521
$347$	− 33.0000i	− 1.77153i	−0.464131	−	0.885766i	$-0.653633\pi$
$347$	− 33.0000i	− 1.77153i	0.464131	−	0.885766i	$-0.346367\pi$
$348$	20.0000i	1.07211i
$349$	7.00000	0.374701	0.187351	−	0.982293i	$-0.440010\pi$
$349$	7.00000	0.374701	0.187351	+	0.982293i	$0.440010\pi$
$350$	0	0
$351$	8.00000	0.427008
$352$	− 4.00000i	− 0.213201i
$353$	− 4.00000i	− 0.212899i	−0.994318	−	0.106449i	$-0.966052\pi$
$353$	− 4.00000i	− 0.212899i	0.994318	−	0.106449i	$-0.0339482\pi$
$354$	22.0000	1.16929
$355$	0	0
$356$	−2.00000	−0.106000
$357$	0	0
$358$	− 12.0000i	− 0.634220i
$359$	26.0000	1.37223	0.686114	−	0.727494i	$-0.259315\pi$
$359$	26.0000	1.37223	0.686114	+	0.727494i	$0.259315\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	7.00000i	0.367912i
$363$	10.0000i	0.524864i
$364$	0	0
$365$	0	0
$366$	4.00000	0.209083
$367$	− 14.0000i	− 0.730794i	−0.930852	−	0.365397i	$-0.880933\pi$
$367$	− 14.0000i	− 0.730794i	0.930852	−	0.365397i	$-0.119067\pi$
$368$	1.00000i	0.0521286i
$369$	−7.00000	−0.364405
$370$	0	0
$371$	0	0
$372$	8.00000i	0.414781i
$373$	22.0000i	1.13912i	0.821951	+	0.569558i	$0.192886\pi$
$373$	22.0000i	1.13912i	−0.821951	+	0.569558i	$0.807114\pi$
$374$	−32.0000	−1.65468
$375$	0	0
$376$	6.00000	0.309426
$377$	20.0000i	1.03005i
$378$	0	0
$379$	−16.0000	−0.821865	−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000	−0.821865	−0.410932	+	0.911666i	$0.634797\pi$
$380$	0	0
$381$	8.00000	0.409852
$382$	19.0000i	0.972125i
$383$	− 29.0000i	− 1.48183i	−0.671598	−	0.740915i	$-0.734392\pi$
$383$	− 29.0000i	− 1.48183i	0.671598	−	0.740915i	$-0.265608\pi$
$384$	−2.00000	−0.102062
$385$	0	0
$386$	20.0000	1.01797
$387$	9.00000i	0.457496i
$388$	− 8.00000i	− 0.406138i
$389$	−32.0000	−1.62246	−0.811232	−	0.584724i	$-0.801203\pi$
$389$	−32.0000	−1.62246	−0.811232	+	0.584724i	$0.801203\pi$
$390$	0	0
$391$	8.00000	0.404577
$392$	7.00000i	0.353553i
$393$	− 24.0000i	− 1.21064i
$394$	−7.00000	−0.352655
$395$	0	0
$396$	4.00000	0.201008
$397$	13.0000i	0.652451i	0.945292	+	0.326226i	$0.105777\pi$
$397$	13.0000i	0.652451i	−0.945292	+	0.326226i	$0.894223\pi$
$398$	1.00000i	0.0501255i
$399$	0	0
$400$	0	0
$401$	−2.00000	−0.0998752	−0.0499376	−	0.998752i	$-0.515902\pi$
$401$	−2.00000	−0.0998752	−0.0499376	+	0.998752i	$0.515902\pi$
$402$	− 4.00000i	− 0.199502i
$403$	8.00000i	0.398508i
$404$	9.00000	0.447767
$405$	0	0
$406$	0	0
$407$	− 4.00000i	− 0.198273i
$408$	16.0000i	0.792118i
$409$	−16.0000	−0.791149	−0.395575	−	0.918434i	$-0.629455\pi$
$409$	−16.0000	−0.791149	−0.395575	+	0.918434i	$0.629455\pi$
$410$	0	0
$411$	−20.0000	−0.986527
$412$	− 17.0000i	− 0.837530i
$413$	0	0
$414$	−1.00000	−0.0491473
$415$	0	0
$416$	−2.00000	−0.0980581
$417$	12.0000i	0.587643i
$418$	− 20.0000i	− 0.978232i
$419$	−8.00000	−0.390826	−0.195413	−	0.980721i	$-0.562605\pi$
$419$	−8.00000	−0.390826	−0.195413	+	0.980721i	$0.562605\pi$
$420$	0	0
$421$	−26.0000	−1.26716	−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000	−1.26716	−0.633581	+	0.773676i	$0.718416\pi$
$422$	22.0000i	1.07094i
$423$	6.00000i	0.291730i
$424$	3.00000	0.145693
$425$	0	0
$426$	28.0000	1.35660
$427$	0	0
$428$	− 10.0000i	− 0.483368i
$429$	16.0000	0.772487
$430$	0	0
$431$	−3.00000	−0.144505	−0.0722525	−	0.997386i	$-0.523019\pi$
$431$	−3.00000	−0.144505	−0.0722525	+	0.997386i	$0.523019\pi$
$432$	4.00000i	0.192450i
$433$	26.0000i	1.24948i	0.780833	+	0.624740i	$0.214795\pi$
$433$	26.0000i	1.24948i	−0.780833	+	0.624740i	$0.785205\pi$
$434$	0	0
$435$	0	0
$436$	−16.0000	−0.766261
$437$	5.00000i	0.239182i
$438$	6.00000i	0.286691i
$439$	−3.00000	−0.143182	−0.0715911	−	0.997434i	$-0.522808\pi$
$439$	−3.00000	−0.143182	−0.0715911	+	0.997434i	$0.522808\pi$
$440$	0	0
$441$	−7.00000	−0.333333
$442$	16.0000i	0.761042i
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	−2.00000	−0.0949158
$445$	0	0
$446$	6.00000	0.284108
$447$	− 6.00000i	− 0.283790i
$448$	0	0
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	0	0
$451$	28.0000	1.31847
$452$	2.00000i	0.0940721i
$453$	− 16.0000i	− 0.751746i
$454$	7.00000	0.328526
$455$	0	0
$456$	−10.0000	−0.468293
$457$	24.0000i	1.12267i	0.827588	+	0.561336i	$0.189713\pi$
$457$	24.0000i	1.12267i	−0.827588	+	0.561336i	$0.810287\pi$
$458$	− 1.00000i	− 0.0467269i
$459$	32.0000	1.49363
$460$	0	0
$461$	24.0000	1.11779	0.558896	−	0.829238i	$-0.311225\pi$
$461$	24.0000	1.11779	0.558896	+	0.829238i	$0.311225\pi$
$462$	0	0
$463$	24.0000i	1.11537i	0.830051	+	0.557687i	$0.188311\pi$
$463$	24.0000i	1.11537i	−0.830051	+	0.557687i	$0.811689\pi$
$464$	−10.0000	−0.464238
$465$	0	0
$466$	−29.0000	−1.34340
$467$	− 36.0000i	− 1.66588i	−0.553362	−	0.832941i	$-0.686655\pi$
$467$	− 36.0000i	− 1.66588i	0.553362	−	0.832941i	$-0.313345\pi$
$468$	− 2.00000i	− 0.0924500i
$469$	0	0
$470$	0	0
$471$	14.0000	0.645086
$472$	11.0000i	0.506316i
$473$	− 36.0000i	− 1.65528i
$474$	22.0000	1.01049
$475$	0	0
$476$	0	0
$477$	3.00000i	0.137361i
$478$	− 13.0000i	− 0.594606i
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	0	0
$481$	−2.00000	−0.0911922
$482$	− 10.0000i	− 0.455488i
$483$	0	0
$484$	−5.00000	−0.227273
$485$	0	0
$486$	−10.0000	−0.453609
$487$	24.0000i	1.08754i	0.839233	+	0.543772i	$0.183004\pi$
$487$	24.0000i	1.08754i	−0.839233	+	0.543772i	$0.816996\pi$
$488$	2.00000i	0.0905357i
$489$	18.0000	0.813988
$490$	0	0
$491$	34.0000	1.53440	0.767199	−	0.641409i	$-0.221650\pi$
$491$	34.0000	1.53440	0.767199	+	0.641409i	$0.221650\pi$
$492$	− 14.0000i	− 0.631169i
$493$	80.0000i	3.60302i
$494$	−10.0000	−0.449921
$495$	0	0
$496$	−4.00000	−0.179605
$497$	0	0
$498$	16.0000i	0.716977i
$499$	29.0000	1.29822	0.649109	−	0.760695i	$-0.275142\pi$
$499$	29.0000	1.29822	0.649109	+	0.760695i	$0.275142\pi$
$500$	0	0
$501$	46.0000	2.05513
$502$	21.0000i	0.937276i
$503$	− 36.0000i	− 1.60516i	−0.596544	−	0.802580i	$-0.703460\pi$
$503$	− 36.0000i	− 1.60516i	0.596544	−	0.802580i	$-0.296540\pi$
$504$	0	0
$505$	0	0
$506$	4.00000	0.177822
$507$	18.0000i	0.799408i
$508$	4.00000i	0.177471i
$509$	11.0000	0.487566	0.243783	−	0.969830i	$-0.421611\pi$
$509$	11.0000	0.487566	0.243783	+	0.969830i	$0.421611\pi$
$510$	0	0
$511$	0	0
$512$	− 1.00000i	− 0.0441942i
$513$	20.0000i	0.883022i
$514$	−8.00000	−0.352865
$515$	0	0
$516$	−18.0000	−0.792406
$517$	− 24.0000i	− 1.05552i
$518$	0	0
$519$	4.00000	0.175581
$520$	0	0
$521$	15.0000	0.657162	0.328581	−	0.944476i	$-0.393430\pi$
$521$	15.0000	0.657162	0.328581	+	0.944476i	$0.393430\pi$
$522$	− 10.0000i	− 0.437688i
$523$	− 20.0000i	− 0.874539i	−0.899331	−	0.437269i	$-0.855946\pi$
$523$	− 20.0000i	− 0.874539i	0.899331	−	0.437269i	$-0.144054\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	14.0000	0.610429
$527$	32.0000i	1.39394i
$528$	8.00000i	0.348155i
$529$	22.0000	0.956522
$530$	0	0
$531$	−11.0000	−0.477359
$532$	0	0
$533$	− 14.0000i	− 0.606407i
$534$	4.00000	0.173097
$535$	0	0
$536$	2.00000	0.0863868
$537$	24.0000i	1.03568i
$538$	31.0000i	1.33650i
$539$	28.0000	1.20605
$540$	0	0
$541$	−30.0000	−1.28980	−0.644900	−	0.764267i	$-0.723101\pi$
$541$	−30.0000	−1.28980	−0.644900	+	0.764267i	$0.723101\pi$
$542$	4.00000i	0.171815i
$543$	− 14.0000i	− 0.600798i
$544$	−8.00000	−0.342997
$545$	0	0
$546$	0	0
$547$	4.00000i	0.171028i	0.996337	+	0.0855138i	$0.0272532\pi$
$547$	4.00000i	0.171028i	−0.996337	+	0.0855138i	$0.972747\pi$
$548$	− 10.0000i	− 0.427179i
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	−50.0000	−2.13007
$552$	− 2.00000i	− 0.0851257i
$553$	0	0
$554$	−10.0000	−0.424859
$555$	0	0
$556$	−6.00000	−0.254457
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	− 4.00000i	− 0.169334i
$559$	−18.0000	−0.761319
$560$	0	0
$561$	64.0000	2.70208
$562$	− 28.0000i	− 1.18111i
$563$	27.0000i	1.13791i	0.822367	+	0.568957i	$0.192653\pi$
$563$	27.0000i	1.13791i	−0.822367	+	0.568957i	$0.807347\pi$
$564$	−12.0000	−0.505291
$565$	0	0
$566$	3.00000	0.126099
$567$	0	0
$568$	14.0000i	0.587427i
$569$	−44.0000	−1.84458	−0.922288	−	0.386503i	$-0.873683\pi$
$569$	−44.0000	−1.84458	−0.922288	+	0.386503i	$0.873683\pi$
$570$	0	0
$571$	−6.00000	−0.251092	−0.125546	−	0.992088i	$-0.540068\pi$
$571$	−6.00000	−0.251092	−0.125546	+	0.992088i	$0.540068\pi$
$572$	8.00000i	0.334497i
$573$	− 38.0000i	− 1.58747i
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	44.0000i	1.83174i	0.401470	+	0.915872i	$0.368499\pi$
$577$	44.0000i	1.83174i	−0.401470	+	0.915872i	$0.631501\pi$
$578$	47.0000i	1.95494i
$579$	−40.0000	−1.66234
$580$	0	0
$581$	0	0
$582$	16.0000i	0.663221i
$583$	− 12.0000i	− 0.496989i
$584$	−3.00000	−0.124141
$585$	0	0
$586$	9.00000	0.371787
$587$	− 27.0000i	− 1.11441i	−0.830375	−	0.557205i	$-0.811874\pi$
$587$	− 27.0000i	− 1.11441i	0.830375	−	0.557205i	$-0.188126\pi$
$588$	− 14.0000i	− 0.577350i
$589$	−20.0000	−0.824086
$590$	0	0
$591$	14.0000	0.575883
$592$	− 1.00000i	− 0.0410997i
$593$	31.0000i	1.27302i	0.771270	+	0.636509i	$0.219622\pi$
$593$	31.0000i	1.27302i	−0.771270	+	0.636509i	$0.780378\pi$
$594$	16.0000	0.656488
$595$	0	0
$596$	3.00000	0.122885
$597$	− 2.00000i	− 0.0818546i
$598$	− 2.00000i	− 0.0817861i
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	−37.0000	−1.50926	−0.754631	−	0.656150i	$-0.772184\pi$
$601$	−37.0000	−1.50926	−0.754631	+	0.656150i	$0.772184\pi$
$602$	0	0
$603$	2.00000i	0.0814463i
$604$	8.00000	0.325515
$605$	0	0
$606$	−18.0000	−0.731200
$607$	8.00000i	0.324710i	0.986732	+	0.162355i	$0.0519090\pi$
$607$	8.00000i	0.324710i	−0.986732	+	0.162355i	$0.948091\pi$
$608$	− 5.00000i	− 0.202777i
$609$	0	0
$610$	0	0
$611$	−12.0000	−0.485468
$612$	− 8.00000i	− 0.323381i
$613$	43.0000i	1.73675i	0.495905	+	0.868377i	$0.334836\pi$
$613$	43.0000i	1.73675i	−0.495905	+	0.868377i	$0.665164\pi$
$614$	12.0000	0.484281
$615$	0	0
$616$	0	0
$617$	− 18.0000i	− 0.724653i	−0.932051	−	0.362326i	$-0.881983\pi$
$617$	− 18.0000i	− 0.724653i	0.932051	−	0.362326i	$-0.118017\pi$
$618$	34.0000i	1.36768i
$619$	10.0000	0.401934	0.200967	−	0.979598i	$-0.435592\pi$
$619$	10.0000	0.401934	0.200967	+	0.979598i	$0.435592\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	− 21.0000i	− 0.842023i
$623$	0	0
$624$	4.00000	0.160128
$625$	0	0
$626$	−10.0000	−0.399680
$627$	40.0000i	1.59745i
$628$	7.00000i	0.279330i
$629$	−8.00000	−0.318981
$630$	0	0
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	11.0000i	0.437557i
$633$	− 44.0000i	− 1.74884i
$634$	27.0000	1.07231
$635$	0	0
$636$	−6.00000	−0.237915
$637$	− 14.0000i	− 0.554700i
$638$	40.0000i	1.58362i
$639$	−14.0000	−0.553831
$640$	0	0
$641$	11.0000	0.434474	0.217237	−	0.976119i	$-0.430296\pi$
$641$	11.0000	0.434474	0.217237	+	0.976119i	$0.430296\pi$
$642$	20.0000i	0.789337i
$643$	− 28.0000i	− 1.10421i	−0.833774	−	0.552106i	$-0.813824\pi$
$643$	− 28.0000i	− 1.10421i	0.833774	−	0.552106i	$-0.186176\pi$
$644$	0	0
$645$	0	0
$646$	−40.0000	−1.57378
$647$	− 7.00000i	− 0.275198i	−0.990488	−	0.137599i	$-0.956061\pi$
$647$	− 7.00000i	− 0.275198i	0.990488	−	0.137599i	$-0.0439386\pi$
$648$	− 11.0000i	− 0.432121i
$649$	44.0000	1.72715
$650$	0	0
$651$	0	0
$652$	9.00000i	0.352467i
$653$	− 30.0000i	− 1.17399i	−0.809590	−	0.586995i	$-0.800311\pi$
$653$	− 30.0000i	− 1.17399i	0.809590	−	0.586995i	$-0.199689\pi$
$654$	32.0000	1.25130
$655$	0	0
$656$	7.00000	0.273304
$657$	− 3.00000i	− 0.117041i
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	4.00000	0.155582	0.0777910	−	0.996970i	$-0.475213\pi$
$661$	4.00000	0.155582	0.0777910	+	0.996970i	$0.475213\pi$
$662$	24.0000i	0.932786i
$663$	− 32.0000i	− 1.24278i
$664$	−8.00000	−0.310460
$665$	0	0
$666$	1.00000	0.0387492
$667$	− 10.0000i	− 0.387202i
$668$	23.0000i	0.889897i
$669$	−12.0000	−0.463947
$670$	0	0
$671$	8.00000	0.308837
$672$	0	0
$673$	19.0000i	0.732396i	0.930537	+	0.366198i	$0.119341\pi$
$673$	19.0000i	0.732396i	−0.930537	+	0.366198i	$0.880659\pi$
$674$	1.00000	0.0385186
$675$	0	0
$676$	−9.00000	−0.346154
$677$	3.00000i	0.115299i	0.998337	+	0.0576497i	$0.0183606\pi$
$677$	3.00000i	0.115299i	−0.998337	+	0.0576497i	$0.981639\pi$
$678$	− 4.00000i	− 0.153619i
$679$	0	0
$680$	0	0
$681$	−14.0000	−0.536481
$682$	16.0000i	0.612672i
$683$	16.0000i	0.612223i	0.951996	+	0.306111i	$0.0990280\pi$
$683$	16.0000i	0.612223i	−0.951996	+	0.306111i	$0.900972\pi$
$684$	5.00000	0.191180
$685$	0	0
$686$	0	0
$687$	2.00000i	0.0763048i
$688$	− 9.00000i	− 0.343122i
$689$	−6.00000	−0.228582
$690$	0	0
$691$	10.0000	0.380418	0.190209	−	0.981744i	$-0.439083\pi$
$691$	10.0000	0.380418	0.190209	+	0.981744i	$0.439083\pi$
$692$	2.00000i	0.0760286i
$693$	0	0
$694$	−33.0000	−1.25266
$695$	0	0
$696$	20.0000	0.758098
$697$	− 56.0000i	− 2.12115i
$698$	− 7.00000i	− 0.264954i
$699$	58.0000	2.19376
$700$	0	0
$701$	−10.0000	−0.377695	−0.188847	−	0.982006i	$-0.560475\pi$
$701$	−10.0000	−0.377695	−0.188847	+	0.982006i	$0.560475\pi$
$702$	− 8.00000i	− 0.301941i
$703$	− 5.00000i	− 0.188579i
$704$	−4.00000	−0.150756
$705$	0	0
$706$	−4.00000	−0.150542
$707$	0	0
$708$	− 22.0000i	− 0.826811i
$709$	−24.0000	−0.901339	−0.450669	−	0.892691i	$-0.648815\pi$
$709$	−24.0000	−0.901339	−0.450669	+	0.892691i	$0.648815\pi$
$710$	0	0
$711$	−11.0000	−0.412532
$712$	2.00000i	0.0749532i
$713$	− 4.00000i	− 0.149801i
$714$	0	0
$715$	0	0
$716$	−12.0000	−0.448461
$717$	26.0000i	0.970988i
$718$	− 26.0000i	− 0.970311i
$719$	−28.0000	−1.04422	−0.522112	−	0.852877i	$-0.674856\pi$
$719$	−28.0000	−1.04422	−0.522112	+	0.852877i	$0.674856\pi$
$720$	0	0
$721$	0	0
$722$	− 6.00000i	− 0.223297i
$723$	20.0000i	0.743808i
$724$	7.00000	0.260153
$725$	0	0
$726$	10.0000	0.371135
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	−72.0000	−2.66302
$732$	− 4.00000i	− 0.147844i
$733$	2.00000i	0.0738717i	0.999318	+	0.0369358i	$0.0117597\pi$
$733$	2.00000i	0.0738717i	−0.999318	+	0.0369358i	$0.988240\pi$
$734$	−14.0000	−0.516749
$735$	0	0
$736$	1.00000	0.0368605
$737$	− 8.00000i	− 0.294684i
$738$	7.00000i	0.257674i
$739$	−18.0000	−0.662141	−0.331070	−	0.943606i	$-0.607410\pi$
$739$	−18.0000	−0.662141	−0.331070	+	0.943606i	$0.607410\pi$
$740$	0	0
$741$	20.0000	0.734718
$742$	0	0
$743$	− 34.0000i	− 1.24734i	−0.781688	−	0.623670i	$-0.785641\pi$
$743$	− 34.0000i	− 1.24734i	0.781688	−	0.623670i	$-0.214359\pi$
$744$	8.00000	0.293294
$745$	0	0
$746$	22.0000	0.805477
$747$	− 8.00000i	− 0.292705i
$748$	32.0000i	1.17004i
$749$	0	0
$750$	0	0
$751$	26.0000	0.948753	0.474377	−	0.880322i	$-0.342673\pi$
$751$	26.0000	0.948753	0.474377	+	0.880322i	$0.342673\pi$
$752$	− 6.00000i	− 0.218797i
$753$	− 42.0000i	− 1.53057i
$754$	20.0000	0.728357
$755$	0	0
$756$	0	0
$757$	8.00000i	0.290765i	0.989376	+	0.145382i	$0.0464413\pi$
$757$	8.00000i	0.290765i	−0.989376	+	0.145382i	$0.953559\pi$
$758$	16.0000i	0.581146i
$759$	−8.00000	−0.290382
$760$	0	0
$761$	13.0000	0.471250	0.235625	−	0.971844i	$-0.424286\pi$
$761$	13.0000	0.471250	0.235625	+	0.971844i	$0.424286\pi$
$762$	− 8.00000i	− 0.289809i
$763$	0	0
$764$	19.0000	0.687396
$765$	0	0
$766$	−29.0000	−1.04781
$767$	− 22.0000i	− 0.794374i
$768$	2.00000i	0.0721688i
$769$	46.0000	1.65880	0.829401	−	0.558653i	$-0.188682\pi$
$769$	46.0000	1.65880	0.829401	+	0.558653i	$0.188682\pi$
$770$	0	0
$771$	16.0000	0.576226
$772$	− 20.0000i	− 0.719816i
$773$	42.0000i	1.51064i	0.655359	+	0.755318i	$0.272517\pi$
$773$	42.0000i	1.51064i	−0.655359	+	0.755318i	$0.727483\pi$
$774$	9.00000	0.323498
$775$	0	0
$776$	−8.00000	−0.287183
$777$	0	0
$778$	32.0000i	1.14726i
$779$	35.0000	1.25401
$780$	0	0
$781$	56.0000	2.00384
$782$	− 8.00000i	− 0.286079i
$783$	− 40.0000i	− 1.42948i
$784$	7.00000	0.250000
$785$	0	0
$786$	−24.0000	−0.856052
$787$	4.00000i	0.142585i	0.997455	+	0.0712923i	$0.0227123\pi$
$787$	4.00000i	0.142585i	−0.997455	+	0.0712923i	$0.977288\pi$
$788$	7.00000i	0.249365i
$789$	−28.0000	−0.996826
$790$	0	0
$791$	0	0
$792$	− 4.00000i	− 0.142134i
$793$	− 4.00000i	− 0.142044i
$794$	13.0000	0.461353
$795$	0	0
$796$	1.00000	0.0354441
$797$	− 32.0000i	− 1.13350i	−0.823890	−	0.566749i	$-0.808201\pi$
$797$	− 32.0000i	− 1.13350i	0.823890	−	0.566749i	$-0.191799\pi$
$798$	0	0
$799$	−48.0000	−1.69812
$800$	0	0
$801$	−2.00000	−0.0706665
$802$	2.00000i	0.0706225i
$803$	12.0000i	0.423471i
$804$	−4.00000	−0.141069
$805$	0	0
$806$	8.00000	0.281788
$807$	− 62.0000i	− 2.18250i
$808$	− 9.00000i	− 0.316619i
$809$	−8.00000	−0.281265	−0.140633	−	0.990062i	$-0.544914\pi$
$809$	−8.00000	−0.281265	−0.140633	+	0.990062i	$0.544914\pi$
$810$	0	0
$811$	44.0000	1.54505	0.772524	−	0.634985i	$-0.218994\pi$
$811$	44.0000	1.54505	0.772524	+	0.634985i	$0.218994\pi$
$812$	0	0
$813$	− 8.00000i	− 0.280572i
$814$	−4.00000	−0.140200
$815$	0	0
$816$	16.0000	0.560112
$817$	− 45.0000i	− 1.57435i
$818$	16.0000i	0.559427i
$819$	0	0
$820$	0	0
$821$	45.0000	1.57051	0.785255	−	0.619172i	$-0.212532\pi$
$821$	45.0000	1.57051	0.785255	+	0.619172i	$0.212532\pi$
$822$	20.0000i	0.697580i
$823$	34.0000i	1.18517i	0.805510	+	0.592583i	$0.201892\pi$
$823$	34.0000i	1.18517i	−0.805510	+	0.592583i	$0.798108\pi$
$824$	−17.0000	−0.592223
$825$	0	0
$826$	0	0
$827$	12.0000i	0.417281i	0.977992	+	0.208640i	$0.0669038\pi$
$827$	12.0000i	0.417281i	−0.977992	+	0.208640i	$0.933096\pi$
$828$	1.00000i	0.0347524i
$829$	16.0000	0.555703	0.277851	−	0.960624i	$-0.410378\pi$
$829$	16.0000	0.555703	0.277851	+	0.960624i	$0.410378\pi$
$830$	0	0
$831$	20.0000	0.693792
$832$	2.00000i	0.0693375i
$833$	− 56.0000i	− 1.94029i
$834$	12.0000	0.415526
$835$	0	0
$836$	−20.0000	−0.691714
$837$	− 16.0000i	− 0.553041i
$838$	8.00000i	0.276355i
$839$	6.00000	0.207143	0.103572	−	0.994622i	$-0.466973\pi$
$839$	6.00000	0.207143	0.103572	+	0.994622i	$0.466973\pi$
$840$	0	0
$841$	71.0000	2.44828
$842$	26.0000i	0.896019i
$843$	56.0000i	1.92874i
$844$	22.0000	0.757271
$845$	0	0
$846$	6.00000	0.206284
$847$	0	0
$848$	− 3.00000i	− 0.103020i
$849$	−6.00000	−0.205919
$850$	0	0
$851$	1.00000	0.0342796
$852$	− 28.0000i	− 0.959264i
$853$	− 30.0000i	− 1.02718i	−0.858036	−	0.513590i	$-0.828315\pi$
$853$	− 30.0000i	− 1.02718i	0.858036	−	0.513590i	$-0.171685\pi$
$854$	0	0
$855$	0	0
$856$	−10.0000	−0.341793
$857$	6.00000i	0.204956i	0.994735	+	0.102478i	$0.0326771\pi$
$857$	6.00000i	0.204956i	−0.994735	+	0.102478i	$0.967323\pi$
$858$	− 16.0000i	− 0.546231i
$859$	−51.0000	−1.74010	−0.870049	−	0.492966i	$-0.835913\pi$
$859$	−51.0000	−1.74010	−0.870049	+	0.492966i	$0.835913\pi$
$860$	0	0
$861$	0	0
$862$	3.00000i	0.102180i
$863$	6.00000i	0.204242i	0.994772	+	0.102121i	$0.0325630\pi$
$863$	6.00000i	0.204242i	−0.994772	+	0.102121i	$0.967437\pi$
$864$	4.00000	0.136083
$865$	0	0
$866$	26.0000	0.883516
$867$	− 94.0000i	− 3.19241i
$868$	0	0
$869$	44.0000	1.49260
$870$	0	0
$871$	−4.00000	−0.135535
$872$	16.0000i	0.541828i
$873$	− 8.00000i	− 0.270759i
$874$	5.00000	0.169128
$875$	0	0
$876$	6.00000	0.202721
$877$	− 27.0000i	− 0.911725i	−0.890050	−	0.455863i	$-0.849331\pi$
$877$	− 27.0000i	− 0.911725i	0.890050	−	0.455863i	$-0.150669\pi$
$878$	3.00000i	0.101245i
$879$	−18.0000	−0.607125
$880$	0	0
$881$	−15.0000	−0.505363	−0.252681	−	0.967550i	$-0.581312\pi$
$881$	−15.0000	−0.505363	−0.252681	+	0.967550i	$0.581312\pi$
$882$	7.00000i	0.235702i
$883$	32.0000i	1.07689i	0.842662	+	0.538443i	$0.180987\pi$
$883$	32.0000i	1.07689i	−0.842662	+	0.538443i	$0.819013\pi$
$884$	16.0000	0.538138
$885$	0	0
$886$	0	0
$887$	− 42.0000i	− 1.41022i	−0.709097	−	0.705111i	$-0.750897\pi$
$887$	− 42.0000i	− 1.41022i	0.709097	−	0.705111i	$-0.249103\pi$
$888$	2.00000i	0.0671156i
$889$	0	0
$890$	0	0
$891$	−44.0000	−1.47406
$892$	− 6.00000i	− 0.200895i
$893$	− 30.0000i	− 1.00391i
$894$	−6.00000	−0.200670
$895$	0	0
$896$	0	0
$897$	4.00000i	0.133556i
$898$	− 30.0000i	− 1.00111i
$899$	40.0000	1.33407
$900$	0	0
$901$	−24.0000	−0.799556
$902$	− 28.0000i	− 0.932298i
$903$	0	0
$904$	2.00000	0.0665190
$905$	0	0
$906$	−16.0000	−0.531564
$907$	3.00000i	0.0996134i	0.998759	+	0.0498067i	$0.0158605\pi$
$907$	3.00000i	0.0996134i	−0.998759	+	0.0498067i	$0.984139\pi$
$908$	− 7.00000i	− 0.232303i
$909$	9.00000	0.298511
$910$	0	0
$911$	5.00000	0.165657	0.0828287	−	0.996564i	$-0.473605\pi$
$911$	5.00000	0.165657	0.0828287	+	0.996564i	$0.473605\pi$
$912$	10.0000i	0.331133i
$913$	32.0000i	1.05905i
$914$	24.0000	0.793849
$915$	0	0
$916$	−1.00000	−0.0330409
$917$	0	0
$918$	− 32.0000i	− 1.05616i
$919$	−24.0000	−0.791687	−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000	−0.791687	−0.395843	+	0.918318i	$0.629548\pi$
$920$	0	0
$921$	−24.0000	−0.790827
$922$	− 24.0000i	− 0.790398i
$923$	− 28.0000i	− 0.921631i
$924$	0	0
$925$	0	0
$926$	24.0000	0.788689
$927$	− 17.0000i	− 0.558353i
$928$	10.0000i	0.328266i
$929$	−34.0000	−1.11550	−0.557752	−	0.830008i	$-0.688336\pi$
$929$	−34.0000	−1.11550	−0.557752	+	0.830008i	$0.688336\pi$
$930$	0	0
$931$	35.0000	1.14708
$932$	29.0000i	0.949927i
$933$	42.0000i	1.37502i
$934$	−36.0000	−1.17796
$935$	0	0
$936$	−2.00000	−0.0653720
$937$	− 38.0000i	− 1.24141i	−0.784046	−	0.620703i	$-0.786847\pi$
$937$	− 38.0000i	− 1.24141i	0.784046	−	0.620703i	$-0.213153\pi$
$938$	0	0
$939$	20.0000	0.652675
$940$	0	0
$941$	6.00000	0.195594	0.0977972	−	0.995206i	$-0.468820\pi$
$941$	6.00000	0.195594	0.0977972	+	0.995206i	$0.468820\pi$
$942$	− 14.0000i	− 0.456145i
$943$	7.00000i	0.227951i
$944$	11.0000	0.358020
$945$	0	0
$946$	−36.0000	−1.17046
$947$	41.0000i	1.33232i	0.745808	+	0.666160i	$0.232063\pi$
$947$	41.0000i	1.33232i	−0.745808	+	0.666160i	$0.767937\pi$
$948$	− 22.0000i	− 0.714527i
$949$	6.00000	0.194768
$950$	0	0
$951$	−54.0000	−1.75107
$952$	0	0
$953$	41.0000i	1.32812i	0.747679	+	0.664060i	$0.231168\pi$
$953$	41.0000i	1.32812i	−0.747679	+	0.664060i	$0.768832\pi$
$954$	3.00000	0.0971286
$955$	0	0
$956$	−13.0000	−0.420450
$957$	− 80.0000i	− 2.58603i
$958$	24.0000i	0.775405i
$959$	0	0
$960$	0	0
$961$	−15.0000	−0.483871
$962$	2.00000i	0.0644826i
$963$	− 10.0000i	− 0.322245i
$964$	−10.0000	−0.322078
$965$	0	0
$966$	0	0
$967$	13.0000i	0.418052i	0.977910	+	0.209026i	$0.0670293\pi$
$967$	13.0000i	0.418052i	−0.977910	+	0.209026i	$0.932971\pi$
$968$	5.00000i	0.160706i
$969$	80.0000	2.56997
$970$	0	0
$971$	20.0000	0.641831	0.320915	−	0.947108i	$-0.396010\pi$
$971$	20.0000	0.641831	0.320915	+	0.947108i	$0.396010\pi$
$972$	10.0000i	0.320750i
$973$	0	0
$974$	24.0000	0.769010
$975$	0	0
$976$	2.00000	0.0640184
$977$	− 62.0000i	− 1.98356i	−0.127971	−	0.991778i	$-0.540847\pi$
$977$	− 62.0000i	− 1.98356i	0.127971	−	0.991778i	$-0.459153\pi$
$978$	− 18.0000i	− 0.575577i
$979$	8.00000	0.255681
$980$	0	0
$981$	−16.0000	−0.510841
$982$	− 34.0000i	− 1.08498i
$983$	− 42.0000i	− 1.33959i	−0.742545	−	0.669796i	$-0.766382\pi$
$983$	− 42.0000i	− 1.33959i	0.742545	−	0.669796i	$-0.233618\pi$
$984$	−14.0000	−0.446304
$985$	0	0
$986$	80.0000	2.54772
$987$	0	0
$988$	10.0000i	0.318142i
$989$	9.00000	0.286183
$990$	0	0
$991$	−23.0000	−0.730619	−0.365310	−	0.930886i	$-0.619037\pi$
$991$	−23.0000	−0.730619	−0.365310	+	0.930886i	$0.619037\pi$
$992$	4.00000i	0.127000i
$993$	− 48.0000i	− 1.52323i
$994$	0	0
$995$	0	0
$996$	16.0000	0.506979
$997$	60.0000i	1.90022i	0.311916	+	0.950110i	$0.399029\pi$
$997$	60.0000i	1.90022i	−0.311916	+	0.950110i	$0.600971\pi$
$998$	− 29.0000i	− 0.917979i
$999$	4.00000	0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1850.2.b.h.149.1		2
5.2	odd	4		1850.2.a.n.1.1	yes	1
5.3	odd	4		1850.2.a.b.1.1	✓	1
5.4	even	2	inner	1850.2.b.h.149.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1850.2.a.b.1.1	✓	1	5.3	odd	4
1850.2.a.n.1.1	yes	1	5.2	odd	4
1850.2.b.h.149.1		2	1.1	even	1	trivial
1850.2.b.h.149.2		2	5.4	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1850.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +2.00000i q^{3} -1.00000 q^{4} +2.00000 q^{6} +1.00000i q^{8} -1.00000 q^{9} +4.00000 q^{11} -2.00000i q^{12} -2.00000i q^{13} +1.00000 q^{16} -8.00000i q^{17} +1.00000i q^{18} +5.00000 q^{19} -4.00000i q^{22} +1.00000i q^{23} -2.00000 q^{24} -2.00000 q^{26} +4.00000i q^{27} -10.0000 q^{29} -4.00000 q^{31} -1.00000i q^{32} +8.00000i q^{33} -8.00000 q^{34} +1.00000 q^{36} -1.00000i q^{37} -5.00000i q^{38} +4.00000 q^{39} +7.00000 q^{41} -9.00000i q^{43} -4.00000 q^{44} +1.00000 q^{46} -6.00000i q^{47} +2.00000i q^{48} +7.00000 q^{49} +16.0000 q^{51} +2.00000i q^{52} -3.00000i q^{53} +4.00000 q^{54} +10.0000i q^{57} +10.0000i q^{58} +11.0000 q^{59} +2.00000 q^{61} +4.00000i q^{62} -1.00000 q^{64} +8.00000 q^{66} -2.00000i q^{67} +8.00000i q^{68} -2.00000 q^{69} +14.0000 q^{71} -1.00000i q^{72} +3.00000i q^{73} -1.00000 q^{74} -5.00000 q^{76} -4.00000i q^{78} +11.0000 q^{79} -11.0000 q^{81} -7.00000i q^{82} +8.00000i q^{83} -9.00000 q^{86} -20.0000i q^{87} +4.00000i q^{88} +2.00000 q^{89} -1.00000i q^{92} -8.00000i q^{93} -6.00000 q^{94} +2.00000 q^{96} +8.00000i q^{97} -7.00000i q^{98} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 4 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 + 4 * q^6 - 2 * q^9	\( 2 q - 2 q^{4} + 4 q^{6} - 2 q^{9} + 8 q^{11} + 2 q^{16} + 10 q^{19} - 4 q^{24} - 4 q^{26} - 20 q^{29} - 8 q^{31} - 16 q^{34} + 2 q^{36} + 8 q^{39} + 14 q^{41} - 8 q^{44} + 2 q^{46} + 14 q^{49} + 32 q^{51} + 8 q^{54} + 22 q^{59} + 4 q^{61} - 2 q^{64} + 16 q^{66} - 4 q^{69} + 28 q^{71} - 2 q^{74} - 10 q^{76} + 22 q^{79} - 22 q^{81} - 18 q^{86} + 4 q^{89} - 12 q^{94} + 4 q^{96} - 8 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 + 4 * q^6 - 2 * q^9 + 8 * q^11 + 2 * q^16 + 10 * q^19 - 4 * q^24 - 4 * q^26 - 20 * q^29 - 8 * q^31 - 16 * q^34 + 2 * q^36 + 8 * q^39 + 14 * q^41 - 8 * q^44 + 2 * q^46 + 14 * q^49 + 32 * q^51 + 8 * q^54 + 22 * q^59 + 4 * q^61 - 2 * q^64 + 16 * q^66 - 4 * q^69 + 28 * q^71 - 2 * q^74 - 10 * q^76 + 22 * q^79 - 22 * q^81 - 18 * q^86 + 4 * q^89 - 12 * q^94 + 4 * q^96 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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